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Secondary Mathematics - Third Year - Geometry

Geometry of Shape and Size
1: Points, Lines, Planes and Angles
  • Demonstrate knowledge and skills related to undefined terms, angles, polygons and circle
    • Describe the ideas of a point, line and plane.
    • Identify, and name the subsets of a line
      • segment
      • ray
    • Name and identify the parts of an angle
    • Determine the measure of an angle using a protractor
    • Illustrate different kinds of angles
      • acute
      • right
      • obtuse
2: Types of Polygons
  • Define, identify and illustrate different kinds of polygons according to the number of sides
    • identify the parts of a regular polygon (vertex angle, central angle, exterior angle)
  • Differentiate convex and non-convex polygons
  • Identify, illustrate and name a triangle
    • its basic parts
    • its secondary parts
  • Classify triangles according to angles and sides
  • Define, illustrate and name a quadrilateral and its parts
  • Identify, illustrate and name the different kinds of quadrilaterals
3: Angles of Polygons
  • Determine the sum of the measures of the angles of polygons
    • the angles of a triangle
    • the exterior angles of a quadrilateral
    • the interior angles of a polygon
  • Define, identify and name the terms related to the circle (radius, diameter and chord)
4: Identifying and measuring Plane and Solid Figures
  • Manifest knowledge and skills in identifying and measuring plane and solid figures and applying these in solving real life problems
    • State and apply the formulas for the measurements of plane and solid figures.
      • perimeter of a triangle, square, and rectangle
      • circumference of a circle
      • area of a triangle, square, parallelogram, trapezoid, and circle
      • surface area of a cube, rectangular prism, square pyramid, cylinder, cone, and a sphere
      • volume of a rectangular prism, triangular prism, pyramid, cylinder, cone, and a sphere
    • solve problems involving plane and solid figures.
Geometric Relations
5: Relations between Segments and Angles
  • Demonstrate knowledge and skills involving relations of segments and angles, sides and angles of a triangle, and solve problems on the relationships between segments and between angles.
    • Define and illustrate betweeness and collinearity of points.
    • Define, identify and illustrate the following
      • congruent segments
      • midpoint of a segment
      • bisector of an angle
  • Solve problems using the definitions and properties involving relationships between segments and between angles.
6: Angle pairs
  • Define, identify and illustrate the different kinds of angle pairs.
    • supplementary
    • complementary
    • adjacent
    • linear pair
    • vertical angles
7: Perpendicularity
  • Define, identify and illustrate perpendicularity.
  • Identify and illustrate the perpendicular bisector of a segment.
8: The Sides and Angles of a Triangle
  • Derive/apply relationships among the sides and angles of a triangle.
    • exterior and corresponding remote interior angles of a triangle
    • triangle inequality
9: Parallel and Transversal Lines
  • Define and illustrate parallel lines.
  • Define and illustrate a transversal.
  • Identify the angles formed by parallel lines cut by a transversal.
  • Determine the relationship between pairs of angles formed by parallel lines cut by a transversal.
    • alternate interior angles
    • alternate exterior angles
    • corresponding angles
    • angles on the same side of the transversal
Triangle Congruence
10: Congruent Triangles
  • Manifest ability to illustrate and apply the conditions for triangle congruence in solving real life problems
    • state and apply the properties of congruence.
      • Reflexive Property
      • Symmetric Property
      • Transitive Property
    • Use inductive skills to prove congruence between triangles.
    • Apply deductive skills to show congruence between triangles.
      • SSS Congruence
      • SAS Congruence
      • ASA Congruence
      • SAA Congruence
11: Congruence and isosceles Triangles
  • Prove congruence properties in an isosceles triangle using the congruence conditions.
    • Congruent sides in a triangle imply that the angles opposite them are congruent
    • Congruent angles in a triangle imply that the sides opposite them are congruent
    • Non-congruent sides in a triangle imply that the angles opposite them are not congruent
    • Non-congruent angles in a triangle imply that the sides opposite them are not congruent
  • Prove inequality properties in an isosceles triangle.
  • Use the conditions triangles congruence to prove congruent segments and congruent angles.
  • Solve routine and non-routine problems.
Properties of Quadrilaterals
12: Trapezoids and Parallelograms
  • Manifest ability to solve practical problems involving types of quadrilaterals and their properties and the conditions that guarantee that a quadrilateral is a parallelogram
    • Apply inductive and deductive skills to derive certain properties of the trapezoid.
      • median of a trapezoid
      • base angles and diagonals of an isosceles trapezoid
    • Apply inductive and deductive skills to derive the properties of a parallelogram
      • each diagonal divides a parallelogram into two congruent triangles
      • opposite angles are congruent
      • non-opposite angles are supplementary
      • opposite sides are congruent
      • diagonals bisect each other
13: Special Quadrilaterials
  • Apply inductive and deductive skills to derive the properties of the diagonals of special quadrilaterals
    • rectangle
    • square
    • rhombus
14: When is a Quadrilateral a Parallelogram?
  • Verify sets of sufficient conditions which guarantee that a quadrilateral is a parallelogram.
  • Apply the conditions to prove that a quadrilateral is a parallelogram.
  • Solve routine and non-routine problems.
Similarity
15: Proportional Segments
  • Demonstrate knowledge and skills in verifying and applying ratio and proportion, proportionality theorems, similarity between triangles and similarities in a right triangle
    • Apply the fundamental law of proportion
      • Product of the means is equal to the product of the extremes
    • Apply the definition of proportional segments to find unknown lengths.
16: Similar Triangles
  • Illustrate and verify the Basic Proportionality Theorem and its converse.
  • Apply the definition of similar triangles.
    • in determining if two triangles are similar
    • in finding the length of a side or measure of an angle of a triangle
  • State and verify the Similarity Theorems.
    • AA similarity
    • SSS similarity
  • Apply the properties of similar triangles and the proportionality theorems to calculate lengths of certain line segments.
  • Apply the AA Similarity Theorem to determine similarities in a right triangle.
    • In a right triangle the altitude to the hypotenuse divides it into two right triangles which are similar to each other and to the given right triangle
17: Pythagorus
  • Apply the definition of similar triangles to derive the Pythagorean Theorem.
    • If a triangle is a right triangle, then the square of the hypotenuse is equal to the sum of the squares of the legs
  • Derive the relationships between the sides of particular triangles using the Pythagorean Theorem.
    • isosceles right triangle
    • 30-60-90 triangle
  • Solve problem involving similar triangles.
Circles
18: Circles and their Properties
  • Demonstrate knowledge and skills related to circles, arcs and angles, tangent lines and tangent circles, and angles formed by tangent and secant lines
    • Define and identify a minor and major arc of a circle.
    • Determine the degree measure of an arc of a circle.
    • Define and identify a central angle.
    • Determine the measure of a central angle.
    • Define and identify an inscribed angle.
    • Determine the measure of an inscribed angle.
    • State and apply the properties of a line tangent to a circle.
      • If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency
      • If two segments from the same exterior point are tangent to a circle, then the two segments are congruent
    • Determine the measure of the angle formed by tangent and/or secant lines.
      • two tangent lines
      • a tangent line and a secant line
      • two secant lines
Plane Coordinate Geometry
19: The Equation of a Line
  • Demonstrate knowledge and skills related to plane coordinate geometry
    • Derive the equation of a line given two points of the line.
    • Determine algebraically the point of intersection of two lines.
    • State and apply the definitions of parallel and perpendicular lines
20: Distance and Midpoint Formulae
  • Derive and state the Distance Formula using the Pythagorean Theorem
  • Derive and state the Midpoint Formula
  • Apply the Distance and Midpoint Formulas to find the lengths of segments and unknown vertices or points.
21: Properties of Triangles and Quadrilaterals using Coordinate Proof
  • Verify properties of triangles and quadrilaterals using coordinate proof.
22: Equation of a Circle
  • Derive/state the standard form of the equation of a circle from the general form.
  • Given the equation of a circle, find its center and radius.
  • Determine the equation of a circle.
    • given its center and radius
    • given its radius and the point of tangency with a given line
  • Solve routine and non-routine problems involving circles.
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